url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
apply le_of_lt
|
case h.left
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x✝⟫_ℝ ≤ 1
|
case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ ⟪α⁻¹ • (LinearIsometryEquiv.symm (InnerProductSpace.toDual ℝ E)) f, x✝⟫_ℝ < 1
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
rw [lt_div_iff hαneg, neg_one_mul, neg_neg]
|
case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ -1 < f x✝ / -α
|
case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ α < f x✝
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
exact hX (by assumption) (by assumption)
|
case h.left.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ α < f x✝
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
assumption
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ E
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
assumption
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
x✝ : E
a✝ : x✝ ∈ X
⊢ x✝ ∈ X
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
rw [div_lt_iff hαneg, neg_one_mul, neg_neg]
|
case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ f x / -α < -1
|
case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ f x < α
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
exact h
|
case h.right
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
x : E
hx : x ∉ X
f : E →L[ℝ] ℝ
α : ℝ
h : f x < α
hX : ∀ b ∈ X, α < f b
hαneg : 0 < -α
⊢ f x < α
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
doublePolarDual_self
|
[135, 1]
|
[159, 7]
|
rw [polarDual_comm]
|
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
hX0 : 0 ∈ X
⊢ X ⊆ polarDual (polarDual X)
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_empty
|
[162, 1]
|
[164, 7]
|
rw [polarDual, Set.preimage_empty, Set.image_empty, Set.image_empty, Set.sInter_empty]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ polarDual ∅ = Set.univ
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_zero
|
[166, 1]
|
[173, 7]
|
rw [polarDual]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ polarDual {0} = Set.univ
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' {0}))) = Set.univ
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_zero
|
[166, 1]
|
[173, 7]
|
have : (@Subtype.val E fun p => p ≠ 0) ⁻¹' {0} = ∅ := by
rw [Set.preimage_singleton_eq_empty]
simp only [ne_eq, Subtype.range_coe_subtype, Set.mem_setOf_eq, not_true, not_false_eq_true]
done
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' {0}))) = Set.univ
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
this : Subtype.val ⁻¹' {0} = ∅
⊢ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' {0}))) = Set.univ
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_zero
|
[166, 1]
|
[173, 7]
|
rw [this, Set.image_empty, Set.image_empty, Set.sInter_empty]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
this : Subtype.val ⁻¹' {0} = ∅
⊢ ⋂₀ (SetLike.coe '' (pointDual '' (Subtype.val ⁻¹' {0}))) = Set.univ
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_zero
|
[166, 1]
|
[173, 7]
|
rw [Set.preimage_singleton_eq_empty]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ Subtype.val ⁻¹' {0} = ∅
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ 0 ∉ Set.range Subtype.val
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_zero
|
[166, 1]
|
[173, 7]
|
simp only [ne_eq, Subtype.range_coe_subtype, Set.mem_setOf_eq, not_true, not_false_eq_true]
|
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
⊢ 0 ∉ Set.range Subtype.val
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
cases' (em (X \ {0}).Nonempty) with hXnonempty hXempty
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
case inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : ¬Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
constructor <;> rw [Metric.isCompact_iff_isClosed_bounded, isBounded_iff_forall_norm_le]
|
case inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) → IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → 0 ∈ interior (polarDual X)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
intro h
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) → IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have : IsOpen (interior (polarDual X)) := isOpen_interior
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : IsOpen (interior (polarDual X))
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [Metric.isOpen_iff] at this
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : IsOpen (interior (polarDual X))
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : ∀ x ∈ interior (polarDual X), ∃ ε > 0, Metric.ball x ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rcases this 0 h with ⟨ ε, hε, hball ⟩
|
case inl.mp
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : ∀ x ∈ interior (polarDual X), ∃ ε > 0, Metric.ball x ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : ∀ x ∈ interior (polarDual X), ∃ ε > 0, Metric.ball x ε ⊆ interior (polarDual X)
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
clear this h
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : 0 ∈ interior (polarDual X)
this : ∀ x ∈ interior (polarDual X), ∃ ε > 0, Metric.ball x ε ⊆ interior (polarDual X)
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
refine ⟨ hXcl, 2/ε, fun x hx => ?_ ⟩
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
⊢ IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
cases' em (x = 0) with hx0 hx0
|
case inl.mp.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : x = 0
⊢ ‖x‖ ≤ 2 / ε
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
let u : E := (ε/2/(norm x)) • x
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have hnormu : ‖u‖ = ε/2 := by
rw [norm_smul, Real.norm_eq_abs, abs_of_pos (div_pos (half_pos hε) (norm_pos_iff.mpr hx0)),
div_mul_cancel _ (norm_ne_zero_iff.mpr hx0)]
done
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have hu : u ∈ Metric.ball (0:E) ε := by
rw [Metric.mem_ball, dist_zero_right, hnormu]
exact half_lt_self hε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have h := interior_subset <| hball hu
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : u ∈ polarDual X
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [mem_polarDual] at h
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : u ∈ polarDual X
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ∀ x ∈ X, ⟪x, u⟫_ℝ ≤ 1
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
specialize h x hx
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ∀ x ∈ X, ⟪x, u⟫_ℝ ≤ 1
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ⟪x, u⟫_ℝ ≤ 1
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [real_inner_smul_right, real_inner_self_eq_norm_mul_norm, ←mul_assoc,
div_mul_cancel _ (norm_ne_zero_iff.mpr hx0), mul_comm, ← div_le_div_right (div_pos hε zero_lt_two),
mul_div_cancel _ (Ne.symm <| ne_of_lt (div_pos hε zero_lt_two)), one_div_div] at h
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ⟪x, u⟫_ℝ ≤ 1
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ‖x‖ ≤ 2 / ε
⊢ ‖x‖ ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact h
|
case inl.mp.intro.intro.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
hu : u ∈ Metric.ball 0 ε
h : ‖x‖ ≤ 2 / ε
⊢ ‖x‖ ≤ 2 / ε
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [hx0, norm_zero]
|
case inl.mp.intro.intro.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : x = 0
⊢ ‖x‖ ≤ 2 / ε
|
case inl.mp.intro.intro.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : x = 0
⊢ 0 ≤ 2 / ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact div_nonneg zero_le_two (le_of_lt hε)
|
case inl.mp.intro.intro.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : x = 0
⊢ 0 ≤ 2 / ε
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [norm_smul, Real.norm_eq_abs, abs_of_pos (div_pos (half_pos hε) (norm_pos_iff.mpr hx0)),
div_mul_cancel _ (norm_ne_zero_iff.mpr hx0)]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
⊢ ‖u‖ = ε / 2
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [Metric.mem_ball, dist_zero_right, hnormu]
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
⊢ u ∈ Metric.ball 0 ε
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
⊢ ε / 2 < ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact half_lt_self hε
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
ε : ℝ
hε : ε > 0
hball : Metric.ball 0 ε ⊆ interior (polarDual X)
x : E
hx : x ∈ X
hx0 : ¬x = 0
u : E := (ε / 2 / ‖x‖) • x
hnormu : ‖u‖ = ε / 2
⊢ ε / 2 < ε
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [interior_eq_compl_closure_compl, Set.mem_compl_iff, Metric.mem_closure_iff, dist_zero_left]
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → 0 ∈ interior (polarDual X)
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → ¬∀ ε > 0, ∃ b ∈ (polarDual X)ᶜ, ‖b‖ < ε
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
push_neg
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → ¬∀ ε > 0, ∃ b ∈ (polarDual X)ᶜ, ‖b‖ < ε
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
intro h
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
⊢ (IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C) → ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
⊢ ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rcases h with ⟨ _, M, hM ⟩
|
case inl.mpr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
h : IsClosed X ∧ ∃ C, ∀ x ∈ X, ‖x‖ ≤ C
⊢ ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
case inl.mpr.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
use 1/M
|
case inl.mpr.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ ∃ ε > 0, ∀ b ∈ (polarDual X)ᶜ, ε ≤ ‖b‖
|
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 1 / M > 0 ∧ ∀ b ∈ (polarDual X)ᶜ, 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
refine ⟨ ?_, ?_ ⟩
|
case h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 1 / M > 0 ∧ ∀ b ∈ (polarDual X)ᶜ, 1 / M ≤ ‖b‖
|
case h.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 1 / M > 0
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ ∀ b ∈ (polarDual X)ᶜ, 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [gt_iff_lt, one_div]
|
case h.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 1 / M > 0
|
case h.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 0 < M⁻¹
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact inv_pos.mpr <| lt_of_lt_of_le (norm_pos_iff.mpr hXnonempty.some_mem.2) (hM hXnonempty.some hXnonempty.some_mem.1)
|
case h.refine_1
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ 0 < M⁻¹
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
intro b hb
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
⊢ ∀ b ∈ (polarDual X)ᶜ, 1 / M ≤ ‖b‖
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : b ∈ (polarDual X)ᶜ
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [Set.mem_compl_iff, mem_polarDual] at hb
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : b ∈ (polarDual X)ᶜ
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : ¬∀ x ∈ X, ⟪x, b⟫_ℝ ≤ 1
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
push_neg at hb
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : ¬∀ x ∈ X, ⟪x, b⟫_ℝ ≤ 1
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : ∃ x ∈ X, 1 < ⟪x, b⟫_ℝ
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rcases hb with ⟨ y, hy, hb ⟩
|
case h.refine_2
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b : E
hb : ∃ x ∈ X, 1 < ⟪x, b⟫_ℝ
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
specialize hM y hy
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
hM : ∀ x ∈ X, ‖x‖ ≤ M
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have hnorminner: |inner y b| ≤ ‖y‖ * ‖b‖ := by
exact abs_real_inner_le_norm y b
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : |⟪y, b⟫_ℝ| ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [abs_of_pos (lt_trans zero_lt_one hb)] at hnorminner
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : |⟪y, b⟫_ℝ| ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have : (1:ℝ) ≤ ‖y‖ * ‖b‖ := le_trans (le_of_lt hb) hnorminner
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
have hynezero: y ≠ 0 := by
rintro rfl
rw [norm_zero, zero_mul] at this
exact not_lt_of_le this zero_lt_one
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : y ≠ 0
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [← norm_pos_iff] at hynezero
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : y ≠ 0
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ 1 / M ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
apply le_trans (div_le_div (le_of_lt <| lt_trans zero_lt_one hb) (le_of_lt hb) hynezero hM)
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ 1 / M ≤ ‖b‖
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ / ‖y‖ ≤ ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
apply div_le_of_nonneg_of_le_mul (le_of_lt hynezero)
|
case h.refine_2.intro.intro
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ / ‖y‖ ≤ ‖b‖
|
case h.refine_2.intro.intro.hc
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ 0 ≤ ‖b‖
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖b‖ * ‖y‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
apply (mul_nonneg_iff_of_pos_left hynezero).mp (le_trans (zero_le_one) this)
|
case h.refine_2.intro.intro.hc
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ 0 ≤ ‖b‖
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖b‖ * ‖y‖
|
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖b‖ * ‖y‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [mul_comm]
|
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖b‖ * ‖y‖
|
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact hnorminner
|
case h.refine_2.intro.intro.h
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
hynezero : 0 < ‖y‖
⊢ ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact abs_real_inner_le_norm y b
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
⊢ |⟪y, b⟫_ℝ| ≤ ‖y‖ * ‖b‖
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rintro rfl
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b y : E
hy : y ∈ X
hb : 1 < ⟪y, b⟫_ℝ
hM : ‖y‖ ≤ M
hnorminner : ⟪y, b⟫_ℝ ≤ ‖y‖ * ‖b‖
this : 1 ≤ ‖y‖ * ‖b‖
⊢ y ≠ 0
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b : E
hy : 0 ∈ X
hb : 1 < ⟪0, b⟫_ℝ
hM : ‖0‖ ≤ M
hnorminner : ⟪0, b⟫_ℝ ≤ ‖0‖ * ‖b‖
this : 1 ≤ ‖0‖ * ‖b‖
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [norm_zero, zero_mul] at this
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b : E
hy : 0 ∈ X
hb : 1 < ⟪0, b⟫_ℝ
hM : ‖0‖ ≤ M
hnorminner : ⟪0, b⟫_ℝ ≤ ‖0‖ * ‖b‖
this : 1 ≤ ‖0‖ * ‖b‖
⊢ False
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b : E
hy : 0 ∈ X
hb : 1 < ⟪0, b⟫_ℝ
hM : ‖0‖ ≤ M
hnorminner : ⟪0, b⟫_ℝ ≤ ‖0‖ * ‖b‖
this : 1 ≤ 0
⊢ False
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact not_lt_of_le this zero_lt_one
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXnonempty : Set.Nonempty (X \ {0})
left✝ : IsClosed X
M : ℝ
b : E
hy : 0 ∈ X
hb : 1 < ⟪0, b⟫_ℝ
hM : ‖0‖ ≤ M
hnorminner : ⟪0, b⟫_ℝ ≤ ‖0‖ * ‖b‖
this : 1 ≤ 0
⊢ False
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [Set.not_nonempty_iff_eq_empty, Set.diff_eq_empty, Set.subset_singleton_iff_eq] at hXempty
|
case inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : ¬Set.Nonempty (X \ {0})
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅ ∨ X = {0}
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
cases' hXempty with hXempty hX0
|
case inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅ ∨ X = {0}
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inr.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
case inr.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hX0 : X = {0}
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [hXempty, polarDual_empty, interior_univ]
|
case inr.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inr.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅
⊢ 0 ∈ Set.univ ↔ IsCompact ∅
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact ⟨ fun _ => isCompact_empty, fun _ => trivial ⟩
|
case inr.inl
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXempty : X = ∅
⊢ 0 ∈ Set.univ ↔ IsCompact ∅
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
rw [hX0, polarDual_zero, interior_univ]
|
case inr.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hX0 : X = {0}
⊢ 0 ∈ interior (polarDual X) ↔ IsCompact X
|
case inr.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hX0 : X = {0}
⊢ 0 ∈ Set.univ ↔ IsCompact {0}
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
compact_polarDual_iff
|
[175, 1]
|
[248, 9]
|
exact ⟨ fun _ => isCompact_singleton, fun _ => trivial ⟩
|
case inr.inr
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hX0 : X = {0}
⊢ 0 ∈ Set.univ ↔ IsCompact {0}
|
no goals
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_compact_if
|
[250, 1]
|
[255, 7]
|
intro h
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
⊢ 0 ∈ interior X → IsCompact (polarDual X)
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
h : 0 ∈ interior X
⊢ IsCompact (polarDual X)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_compact_if
|
[250, 1]
|
[255, 7]
|
rw [← doublePolarDual_self hXcl hXcv (interior_subset h), compact_polarDual_iff (polarDual_closed _)] at h
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
h : 0 ∈ interior X
⊢ IsCompact (polarDual X)
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
h : IsCompact (polarDual X)
⊢ IsCompact (polarDual X)
|
https://github.com/Jun2M/Main-theorem-of-polytopes.git
|
fb84f7409e05ca9db3a1bbfcd4d0a16001515fe8
|
src/Polar.lean
|
polarDual_compact_if
|
[250, 1]
|
[255, 7]
|
exact h
|
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : FiniteDimensional ℝ E
X : Set E
hXcl : IsClosed X
hXcv : Convex ℝ X
h : IsCompact (polarDual X)
⊢ IsCompact (polarDual X)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_pos
|
[180, 1]
|
[182, 25]
|
unfold hminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < hminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_pos
|
[180, 1]
|
[182, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_le_hmaxₚ
|
[184, 1]
|
[185, 31]
|
unfold hminₚ hmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ hminₚ ≤ hmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_le_hmaxₚ
|
[184, 1]
|
[185, 31]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_pos
|
[193, 1]
|
[195, 25]
|
unfold wminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < wminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_pos
|
[193, 1]
|
[195, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_le_wmaxₚ
|
[197, 1]
|
[198, 31]
|
unfold wminₚ wmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ wminₚ ≤ wmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_le_wmaxₚ
|
[197, 1]
|
[198, 31]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.Rmaxₚ_pos
|
[203, 1]
|
[205, 25]
|
unfold Rmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < Rmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.Rmaxₚ_pos
|
[203, 1]
|
[205, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.σₚ_pos
|
[210, 1]
|
[212, 22]
|
unfold σₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < σₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.σₚ_pos
|
[210, 1]
|
[212, 22]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₁ₚ_pos
|
[217, 1]
|
[219, 23]
|
unfold F₁ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₁ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₁ₚ_pos
|
[217, 1]
|
[219, 23]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₂ₚ_pos
|
[224, 1]
|
[226, 23]
|
unfold F₂ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₂ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₂ₚ_pos
|
[224, 1]
|
[226, 23]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
gaussianPdf_pos
|
[24, 1]
|
[27, 53]
|
refine' mul_pos (div_pos zero_lt_one (sqrt_pos.2 (mul_pos _ h.det_pos))) (exp_pos _)
|
n : ℕ
R : Matrix (Fin n) (Fin n) ℝ
y : Fin n → ℝ
h : R.PosDef
⊢ 0 < gaussianPdf R y
|
n : ℕ
R : Matrix (Fin n) (Fin n) ℝ
y : Fin n → ℝ
h : R.PosDef
⊢ 0 < (2 * π) ^ n
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
gaussianPdf_pos
|
[24, 1]
|
[27, 53]
|
exact (pow_pos (mul_pos (by positivity) pi_pos) n)
|
n : ℕ
R : Matrix (Fin n) (Fin n) ℝ
y : Fin n → ℝ
h : R.PosDef
⊢ 0 < (2 * π) ^ n
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
gaussianPdf_pos
|
[24, 1]
|
[27, 53]
|
positivity
|
n : ℕ
R : Matrix (Fin n) (Fin n) ℝ
y : Fin n → ℝ
h : R.PosDef
⊢ 0 < 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
have : ∀ i,
i ∈ Finset.univ → gaussianPdf R (y i) ≠ 0 := fun i _ => ne_of_gt (gaussianPdf_pos _ _ hR)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
⊢ (∏ i : Fin N, gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
⊢ (∏ i : Fin N, gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
have sqrt_2_pi_n_R_det_ne_zero: sqrt ((2 * π) ^ n * R.det) ≠ 0 := by
refine' ne_of_gt (sqrt_pos.2 (mul_pos _ hR.det_pos))
exact (pow_pos (mul_pos (by positivity) pi_pos) _)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
⊢ (∏ i : Fin N, gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ (∏ i : Fin N, gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
rw [log_prod Finset.univ (fun i => gaussianPdf R (y i)) this]
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ (∏ i : Fin N, gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ ∑ i : Fin N, (gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
unfold gaussianPdf
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ ∑ i : Fin N, (gaussianPdf R (y i)).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ ∑ i : Fin N, (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
apply congr_arg (Finset.sum Finset.univ)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ ∑ i : Fin N, (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log =
∑ i : Fin N, (-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2)
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ (fun i => (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log) = fun i =>
-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
ext i
|
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
⊢ (fun i => (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log) = fun i =>
-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2
|
case h
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log =
-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
rw [log_mul, log_div, sqrt_mul, log_mul, log_exp, log_one, zero_sub]
|
case h
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (1 / ((2 * π) ^ n * R.det).sqrt * (-R⁻¹.quadForm (y i) / 2).exp).log =
-(((2 * π) ^ n).sqrt.log + R.det.sqrt.log) + -R⁻¹.quadForm (y i) / 2
|
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ R.det.sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 0 ≤ (2 * π) ^ n
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 / ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (-R⁻¹.quadForm (y i) / 2).exp ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
simp [rpow_eq_pow]
|
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ R.det.sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 0 ≤ (2 * π) ^ n
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 / ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (-R⁻¹.quadForm (y i) / 2).exp ≠ 0
|
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ¬((2 * π) ^ n).sqrt = 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ R.det.sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 0 ≤ (2 * π) ^ n
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 / ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (-R⁻¹.quadForm (y i) / 2).exp ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/CovarianceEstimation.lean
|
log_prod_gaussianPdf
|
[33, 1]
|
[53, 22]
|
exact ne_of_gt (sqrt_pos.2 (pow_pos (mul_pos (by positivity) pi_pos) _))
|
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ¬((2 * π) ^ n).sqrt = 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ R.det.sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 0 ≤ (2 * π) ^ n
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 / ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (-R⁻¹.quadForm (y i) / 2).exp ≠ 0
|
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ R.det.sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 0 ≤ (2 * π) ^ n
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hx
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ 1 / ((2 * π) ^ n * R.det).sqrt ≠ 0
case h.hy
N n : ℕ
y : Fin N → Fin n → ℝ
R : Matrix (Fin n) (Fin n) ℝ
hR : R.PosDef
this : ∀ i ∈ Finset.univ, gaussianPdf R (y i) ≠ 0
sqrt_2_pi_n_R_det_ne_zero : ((2 * π) ^ n * R.det).sqrt ≠ 0
i : Fin N
⊢ (-R⁻¹.quadForm (y i) / 2).exp ≠ 0
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.